Renormalization of the meson-baryon SU(3) Lagrangian in chiral perturbation theory to order

L(M2)
=
F02 4
(<
∂µU ∂µU †
>
+2B0
<
M(U
+
U †)
>)
(3)
ae-mail: mueller@pythia.itkp.uni-bonn.de
1
The scale F0 can be identified with the pseudoscalar decay constant in the chiral limit and the constant B0 is related to the quark condensate. One can now calculate tree diagrams using the effective Lagrangian LM and derive all current algebra predictions. One has to include higher order corrections
factor Λ is given by
µd−4 Λ = (4π)2
d
1 −
4
−
1 2
[log(4π)
+
1
+
Γ′(1)]
(7)
depending on the arbitrary renormalization scale µ. The sum of one-loop amplitudes and of tree amplitudes based on (5) is not only finite, but also independent of the scale µ. To evaluate these loop graphs one considers the neighbourhood of the solution to the classical equations of motion. The contribution to the one-loop graphs to the generating functional is given by
and meson loops, because tree diagrams are always real and thus unitarity is violated. Weinberg made the important observation that diagrams with L (L = 1, 2, . . .) meson loops are suppressed by powers of (q2)L with respect to the leading term. In the meson sector we have a one–to–one correspondence
Li = Lri (µ) + ΓiΛ .
(6)
The constants Γi serve to renormalize the infinities of the meson loops and the remaining finite pieces Lri (µ) have to be fixed phenomenologically or to be estimated by some model. With dimensional regularization, the divergent
of LM where the chiral expansion proceeds in steps of two powers of q. So we
have
LMB = L(M1)B + L(M2)B + L(M3)B + . . . .
(9)
The effective meson–baryon Lagrangian for processes with one incoming and one outgoing baryon is bilinear in the baryon field and starts with terms of
exp{iZ1−loop}
=
N
[det(dd
+
σ)]−
1 2
(8)
2
where the explicit definitions are given in 4. The determinant is UV divergent
and produces poles at d = 0, 2, 4, . . . in dimensional regularization and can be regularized via the heat kernel expansion.
between the number L of loops and the chiral dimension
D = 2 + Nd(d − 2) + 2L
(4)
d
where Nd denotes the number of vertices of dimension d. For a given Smatrix element, the chiral dimension D increases with the number of meson
noted by D and F and the average baryon octet mass m in the chiral limit.
10
2
L4 = LiPi + HiP˜i
(5)
i=1
i=1
The rLeabharlann normalization procedure at one-loop level is then achieved by simply dividing the coefficients in (5) into a scale dependent finite and divergent part:
loops. It should be not surprising that a generic one-loop diagram in CHPT is
divergent. The divergences can be taken care of once and for all by calculating the divergent part of the one-loop functional. Since it must be a local action with the symmetries of L(M2) and since it has chiral dimension D = 4, the corresponding Lagrangian must be of the general form as discussed in ref 4.
Let us now consider the meson-baryon interaction in CHPT. Because of
the different Lorentz structure of meson and baryon fields, the chiral expansion
of LMB contains terms of O(qn) for each positive integer n, unlike in the case
dimension one,
L(M1)B
=<
B¯
(i∇/ −
m) B
>
+D 2
<
B¯ {u/γ5, B}
>
+F 2
< B¯ [u/γ5, B] >
. (10)
We
collect
the
spin–
1 2
baryon octet in
the
3×3
matrix
B.
The
lowest order
meson–baryon Lagrangian contains two axial–vector coupling constants, de-
Nussallee 14 - 16 53115 Bonn, Germany
We perform the complete regularization of all Green functions with a single incoming and outgoing baryon to order q3 in the low energy expansion for the SU(3) meson-baryon system in chiral perturbation theory. The divergences can be extracted in a chiral invariant manner by making use of the heat kernel representation of the propagators in d-dimensional Euclidean space.
LM = L(M2) + L(M4) + . . .
(1)
where the index (d = 2, 4, . . .) denotes the low energy dimension. The leading term in the low energy expansion (1) can be easily written down in terms of the pseudoscalar Goldstone fields (φ = π, K, η) which are collected in a 3 × 3 unimodular, unitary matrix U (x),